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Hilbert transform is a quite sensitive topic here, since Gabor's paper, Theory of Communication, J. Inst. Electr. Engineering, London, 1946. Perhaps even more important than the Fourier transform.

There are short Gaussian FIR approximations, smoothing filters, or polynomial fits.

What would be methods for very short (say, less than $16$ taps) Hilbert tranform approximations of a signal, possibly including a weighting factor? I am intending to use them on small chunks of signals, in moving frames that progress along data as new samples are acquired. Signals can be considered freed from a potential low-frequency trend (e.g. polynomial).

Fey features would be:

  • a management or reduction of small-sized frame artifacts (I am aware of those for short Fourier frames, much less with short Hilbert transforms),
  • potential asymmetric windowing to add a forgeting factor to older samples,
  • a recursive formulation, in the spirit of the sliding DFT, would be a plus.

Finally, an IIR formulation, such as the one of the recursive least squares (RLS) adaptive filters, or the exponentially-weighted moving average filter, would be very interesting as well.

Royi
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Laurent Duval
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  • Off topic: It' not proper to compare Hilbert transform to the Fourier transform, eventhough both share the title "transform". The Fourier is the most widely used tool to "analyse" the character of signals & systems in many branches of physical sciences and engineering, whereas the Hilbert is not used for analysis of any sort, but is a "tool" to generate an "Analytic signal" which is useful for the solution of some problems. Consider the fact that it's the Fourier transform which is used to analyse the characteristic of the very Hilbert transform itself and not vice versa... – Fat32 Apr 02 '16 at 22:28
  • @Fat32 I am not sure about what you call "off topic". I am not comparing Fourier and Hilbert only by their "transform" aspect. I strongly believe they have stronger connections that (some) other orthogonal transforms, and that they help understand each other (to me at list, with local versions for local phase), and that Hilbert transform might be more fundamental. – Laurent Duval Apr 02 '16 at 22:41
  • @Fat32 by the way, are you aware of the precursor of the Dirac distribution in the original work by Fourier? – Laurent Duval Apr 02 '16 at 22:44
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    off topic was my comment itself, as it was targetted not to your question but to a link inside it. I just take it general as for anyone who compares them, and not you, as you clearly didn't. And for the deep connections, Hilbert transform seems to have a relation to complex signals, analytic function theory etc. They might belong to some category of transforms but their use is almost completely distinct of each other, to my knowledge. – Fat32 Apr 02 '16 at 22:46
  • what do you mean by "precursor of the Dirac distribution" ? – Fat32 Apr 02 '16 at 22:51
  • Probably off-topic too, just crossed my mind on how topics collide. You can find a precursor (as a cardinal sine) of the Dirac distribution, as a neutral element to the convolution, in his book – Laurent Duval Apr 02 '16 at 23:00
  • Hmm actually I havent seen any of his original works, all I can say is interesting. Also science historians have tendency to relate every invention back to antiquities, so I wouldn't be surprised if they said Egyptians also used dirac distributions :)) On the other hand strange enough as a cross of mind, recently I was thinking about "imaginary" numbers and I said, what would science look like without imaginary numbers? A feeling whispered as if everything should be the same, as they dont exist really ? so can a nonexisting thing have a positive effect on reality ? – Fat32 Apr 02 '16 at 23:09
  • Let us continue this discussion in chat. – Laurent Duval Apr 02 '16 at 23:20
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    I don't understand the last part of the last sentence: "... (removed from a trend to keep it sufficiently harmonic), possibly including a weighting factor". Apart from that, if you don't want an answer suggesting to just take a 16 tap windowed version of the impulse response of an ideal Hilbert transformer, you need to explain why this isn't good enough and what your criteria are for judging if a Hilbert transformer is "good" or not. – Matt L. Apr 03 '16 at 09:52
  • @Matt L. I have edited the OP based on your comments, hoping the context has becoome clearer – Laurent Duval Apr 03 '16 at 15:58

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This is more like an extended comment to chart the possible answers.

Hilbert transform is a frequency domain 90-degree phase shift of the signal. It has an antisymmetrical impulse response around time = 0. You specify that the approximation shall be causal (EDIT: this requirement has since been removed), so I think you need to reference the phase shift to that provided by another filter. If we denote the two filters F1 and F2, some usable alternatives are:

Case | Filter F1    | Filter F2           | Phase diff     | Magnitude frequency responses
-----+--------------+---------------------+----------------+------------------------------
  C1 | pure delay   | antisymmetrical FIR | exactly 90 deg | F2 has ripple
  C2 | pure delay   | all-pass IIR        | approx 90 deg  | No ripple
  C3 | all-pass IIR | all-pass IIR        | approx 90 deg  | No ripple

In all those cases you have two filters whose phase frequency responses have a phase difference of exactly or close to 90 degrees over the desired frequency band.

If the filter outputs are used "in quadrature" (summed, with the other multiplied by the imaginary unit) to create an analytic signal, any phase frequency response ripple will become magnitude frequency response ripple of the composite negative frequency removal filter.

Olli Niemitalo
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